Time Series analysis for Facebook message volume

Choosing a benchmark model

Two possible benchmark models would be the naive model and the mean forecast. Unsurprisingly, both of the models have unacceptable residuals, with significant autocorrelations in the ACF chart, and the Ljung-Box test rejecting the null hypothesis that they are not correlated.

naiveModel <- naive(train.ts,h = 12)
meanModel <- meanf(train.ts,h = 12)

The graphical analyses of the residuals were omitted from the report, however both show substantial autocorrelation and the distribution of the residuals do not appear normal, as judged both visually and through the Ljung-Box test.

Arima family

mArima1<- auto.arima(train.ts)
fa1 <- forecast(mArima1,h = 12)
checkresiduals(mArima1)

  Ljung-Box test

 data:  Residuals from ARIMA(1,0,1) with zero mean
 Q* = 19.786, df = 22, p-value = 0.5965

 Model df: 2.   Total lags used: 24

Using the auto.Arima() function, I fitted an ARIMA model. This yields an ARIMA(1,0,1) model, which means that it is the combination of a first order autoregressive model (\(\phi\) = 0.98) with no differencing and a first order moving average (\(\theta\) = -0.57). This model seems reasonable given the sample ACF and PACF plots.

I considered adding a constant to the model, however that would lead to an overestimation of the long term trend of message volume. The zero mean assumption makes sense in that the limit of \(\widehat{y}_{t+h|t}\) would approach a constant determined by the last few observations.

The autocorrelation seems appropriate, although the residuals do diverge visually from normal in QQ-plots. Nonetheless, given the amount of data we have I will accept this model.

ETS Family

etsModel1 <- ets(train.ts)
checkresiduals(etsModel1)+theme_tufte()

##
  Ljung-Box test
data:  Residuals from ETS(A,N,N)
Q* = 19.59, df = 22, p-value = 0.6087
Model df: 2.   Total lags used: 24

## NULL

fe1 <- forecast(etsModel1,h = 12)

The model selected is \(ETS(A,N,N)\). The smoothing parameter is \(\alpha=\) 0.42. This means that the growth, seasonal states do not change from their initial values. There is no damping parameter. Again, the residuals do not seem to be perfectly normally distributted, but this is not entirely surprising given the shortness of the time series. I will be content with the fact that they are not significantly non-normally distributed.

In order to select a model from these two I use time series cross validation following Hyndman’s example.

fets <- function(x, h) {
  forecast(ets(x), h = h)
}
farima <- function(x, h) {
forecast(auto.arima(x), h=h)
}
e1 <- tsCV(train.ts, fets, h=1)#Compute CV errors for ETS as e1
e2 <- tsCV(train.ts, farima, h=1)#Compute CV errors for ARIMA as e2
Find MSE of each model class
mean(e1^2, na.rm=TRUE)

## [1] 10710.59

mean(e2^2, na.rm=TRUE)

## [1] 14473.21

Given that the MSE for the ETS model is lower, I would choose to select that to forecast.